A formal note · IV

Stochastic memetic replication and the categorical replicator–mutator equation

Selection, mutation, semantic coarse-graining, and evolutionary dynamics in an ∞-topos.

probability monad / fitness / Markov kernel / transfer operator / lumpability

Abstract

We extend the ∞-topos and network-theoretic formulation of memetic replication from deterministic transformations to stochastic population dynamics.

Let

H = Sh_{\infty} (C, J)

be a cognitive semantic ∞-topos, and let $S \in H$ be an object of semantic memetic classes. Since probability is not canonically determined by an arbitrary ∞-topos, we assume explicitly that $H$ is equipped with a commutative probability monad

P : H \to H

or, equivalently for the present purposes, with a compatible Markov-category structure.

A population is a probability state

μ : 1 \to P (S) .

Selection is specified by a positive fitness observable

w : S \to R_{> 0},

while mutation and imperfect transmission are represented by a Markov kernel

K : S \to P (S) .

The resulting discrete-time evolutionary operator is

R_{w, K} (μ) = K_{*} \frac{w μ}{\int _{S} w d μ} .

Equivalently, introducing the positive transfer operator $L_{w, K} μ = K_{*} (w μ)$ , one has

R_{w, K} (μ) = \frac{L _{w, K} μ}{L _{w, K} μ ( S )} .

For a finite semantic state space this becomes the classical replicator–mutator equation

x_{j}^{'} = \frac{i \sum x _{i} w _{i} Q _{ij}}{i \sum x _{i} w _{i}} .

We prove normalization, characterize stationary populations as positive eigenmeasures of the transfer operator, derive both continuous-time replicator and mutation–selection equations, establish a coarse-graining theorem showing precisely when observer-relative semantic quotients inherit closed Markovian dynamics, and derive the Perron–Frobenius asymptotics of finite primitive systems.

The resulting framework identifies stochastic memetic evolution with normalized positive dynamics on capability-relative semantic classes. The deterministic theories are Note I, Note II, and Note III.

Notation

The symbols of this note, in the order they become active.

$S$: Semantic state object: the distinctions an observer retains.
$P$: A commutative probability monad on the cognitive ∞-topos.
$μ : 1 \to P (S)$: A population: a probability state on semantic classes.
$w$: Positive fitness. Mean fitness is the integral of w against μ.
$K$: A Markov kernel. Mutation, noise, and imperfect transmission.
$Sel_{w}$: Selection: reweight μ by fitness and renormalize.
$R_{w, K}$: The replicator–mutator operator, selection followed by mutation.
$L_{w, K}$: The positive transfer operator. Evolution is its projectivization.
$Q_{ij}$: Finite mutation matrix. Rows are probability distributions.
$q$: A coarse-graining. Dynamics on the quotient closes only when it is lumpable.

1Semantic state space

Let $(C, J)$ be a cognitive ∞-site and let

H = Sh_{\infty} (C, J) .

Let $M \in H$ be an underlying object of memetic realizations, and let $Q : M \to S$ be a semantic observable, localization, or quotient. The object $S \in H$ is the semantic state object.

Two realizations $m, m^{'} \in M$ are equivalent relative to $Q$ when $Q (m) ≃ Q (m^{'})$ . Thus $S$ represents the memetic distinctions retained by the observer or cognitive architecture under consideration.

2Probability structure

Probability is additional structure. An arbitrary ∞-topos does not by itself determine a canonical notion of probability measure. We therefore assume the following.

Remark2.1Probability monad

The ∞-topos $H$ is equipped with a commutative probability monad

P : H \to H

with unit

η_{X} : X \to P (X)

and multiplication

m_{X} : P (P (X)) \to P (X) .

We assume sufficient integration structure to make expressions of the form

\int_{X} f d μ

well defined for suitable internal nonnegative real-valued functions $f : X \to R_{\geq 0}$ . In ordinary measurable spaces one may take the Giry probability monad. For finite objects, $P (X)$ is the probability simplex on $X$ .

3Markov kernels

Definition3.1Stochastic memetic transformation

A stochastic transformation from $X$ to $Y$ is a Kleisli morphism

K : X ⟶ P (Y) .

We call such a morphism a Markov kernel. For $x \in X$ , the probability state $K (x) \in P (Y)$ describes the distribution of possible outputs conditional on input $x$ .

Definition3.2Composition

If $K : X \to P (Y)$ and $L : Y \to P (Z)$ , their Kleisli composite

L ⊙ K : X \to P (Z)

is defined by

m_{Z} \circ P (L) \circ K .

In ordinary integral notation,

(L ⊙ K) (x) = \int_{Y} L (y) K (x, d y) .

Thus stochastic transmission is closed under composition.

4Population states

Definition4.1Population distribution

A memetic population state on $S$ is a probability state

μ : 1 \to P (S) .

The space of populations is therefore $P (S)$ . For finite $S = {1, \dots, n}$ , we identify $μ$ with a vector $x = (x_{1}, \dots, x_{n})$ satisfying

x_{i} \geq 0, i \sum x_{i} = 1.

5Fitness

Definition5.1Fitness observable

A fitness function is a positive semantic observable

w : S \to R_{> 0} .

For a state $s$ , the value $w (s)$ is its expected reproductive weight. The population mean fitness is

w ˉ_{μ} = \int_{S} w (s) d μ (s) .

We assume $0 < w ˉ_{μ} < \infty$ for every population under consideration.

6Selection

Definition6.1Selection operator

The fitness-weighted population $Sel_{w} (μ)$ is the probability distribution defined by

d Sel_{w} (μ) (s) = \frac{w ( s )}{w ˉ _{μ}} d μ (s) .

Equivalently, for an event $A \subseteq S$ ,

Sel_{w} (μ) (A) = \frac{\int _{A} w d μ}{\int _{S} w d μ} .

Proposition6.2Selection preserves normalization

Sel_{w} (μ) (S) = 1.

Proof.

By definition the numerator and the denominator are the same integral:

Sel_{w} (μ) (S) = \frac{\int _{S} w d μ}{\int _{S} w d μ} = 1.

7Mutation and imperfect transmission

Definition7.1Mutation kernel

A mutation kernel is a Markov kernel

K : S \to P (S) .

For $s \in S$ , the distribution $K (s)$ describes the possible semantic type of an offspring or transmitted realization originating from $s$ . Mutation here includes any stochastic failure of exact semantic inheritance:

copying errors
reinterpretation
translation
recombination noise

or any other probabilistic change of semantic class.

8Pushforward through a Markov kernel

A kernel induces a map on populations

K_{*} : P (S) \to P (S),

given by

(K_{*} μ) (A) = \int_{S} K (s, A) d μ (s) .

Proposition8.1Mutation preserves probability mass

If $μ$ is normalized, then $K_{*} μ$ is normalized.

Proof.

Since each $K (s)$ is a probability measure, $K (s, S) = 1$ . Hence

(K_{*} μ) (S) = \int_{S} 1 d μ (s) = 1.

9The categorical replicator–mutator operator

We now compose selection and mutation.

Definition9.1Replicator–mutator operator

Define

R_{w, K} = K_{*} \circ Sel_{w} .

Thus

R_{w, K} (μ) = K_{*} \frac{w μ}{\int _{S} w d μ} .

Explicitly, for an event $A$ ,

R_{w, K} (μ) (A) = \frac{\int _{S} w ( s ) K ( s , A ) d μ ( s )}{\int _{S} w ( s ) d μ ( s )} .

10Normalization theorem

Theorem10.1The operator preserves probability

For every probability distribution $μ$ , $R_{w, K} (μ)$ is again a probability distribution.

Proof.

Selection produces a probability state by Proposition 6.2. The Markov pushforward $K_{*}$ preserves probability mass by Proposition 8.1. Therefore their composite does also.

Thus

R_{w, K} : P (S) \to P (S)

defines a dynamical system on the population simplex.

11The positive transfer operator

Normalization hides a useful linear structure. Let $M_{+} (S)$ denote the cone of finite nonnegative measures on $S$ .

Definition11.1Mutation–selection transfer operator

Define

L_{w, K} : M_{+} (S) \to M_{+} (S)

(L_{w, K} ν) (A) = \int_{S} w (s) K (s, A) d ν (s) .

Equivalently, $L_{w, K} = K_{*} \circ M_{w}$ , where $M_{w} (ν) = w ν$ . The operator is positive and linear on finite measures.

Proposition11.2Total mass is mean fitness

For a probability measure $μ$ , $L_{w, K} μ (S) = w ˉ_{μ}$ .

Proof.

L_{w, K} μ (S) = \int_{S} w (s) K (s, S) d μ (s) = \int_{S} w (s) d μ (s) = w ˉ_{μ} .

Corollary11.3Projectivization

The replicator–mutator operator is the projectivization of the positive transfer operator:

R_{w, K} (μ) = \frac{L _{w, K} μ}{L _{w, K} μ ( S )} .

This is the basic categorical structure behind the theory. Evolution takes place in the projectivization of a positive cone.

12Finite-state form

Let $S = {1, \dots, n}$ . Write $x_{i} = μ ({i})$ , let $w_{i} > 0$ , and define $Q_{ij} = K (i, {j})$ . The matrix $Q$ is row-stochastic:

Q_{ij} \geq 0, j \sum Q_{ij} = 1.

Theorem12.1Finite replicator–mutator equation

The categorical operator reduces to

x_{j}^{'} = \frac{i \sum x _{i} w _{i} Q _{ij}}{i \sum x _{i} w _{i}} .

Proof.

By Definition 9.1, the mass of ${j}$ is

R_{w, K} (μ) ({j}) = \frac{i \sum w _{i} K ( i , { j }) x _{i}}{i \sum w _{i} x _{i}} .

Substituting $K (i, {j}) = Q_{ij}$ gives the result.

In row-vector notation,

x^{'} = \frac{x W Q}{x W 1},

where $W = diag (w_{1}, \dots, w_{n})$ . Since $Q 1 = 1$ , this becomes

x^{'} = \frac{x A}{x A 1}, A = W Q .

13Pure replication

Set $K = id$ . Then $Q = I$ , and the update becomes

x_{i}^{'} = \frac{x _{i} w _{i}}{w ˉ},

where $w ˉ = \sum_{i} x_{i} w_{i}$ . This is the discrete replicator equation.

14Pure mutation

If fitness is constant, $w (s) = c > 0$ , then selection is trivial: $Sel_{w} (μ) = μ$ . Hence

μ^{'} = K_{*} μ .

For a finite state space, $x^{'} = x Q$ . Ordinary Markov dynamics is the zero-selection limit of the memetic evolutionary theory.

15Stationary populations

Definition15.1Evolutionary equilibrium

A population $μ_{*} \in P (S)$ is stationary if

R_{w, K} (μ_{*}) = μ_{*} .

16Eigenmeasure theorem

Theorem16.1Equilibria are normalized eigenmeasures

A probability distribution $μ_{*}$ is stationary if and only if

L_{w, K} μ_{*} = λ μ_{*}

for $λ = \int_{S} w d μ_{*}$ .

Proof.

Suppose first that $R_{w, K} (μ_{*}) = μ_{*}$ . By Corollary 11.3,

μ_{*} = \frac{L _{w, K} μ _{*}}{L _{w, K} μ _{*} ( S )} .

Hence $L_{w, K} μ_{*} = L_{w, K} μ_{*} (S) μ_{*}$ . By Proposition 11.2, $L_{w, K} μ_{*} (S) = \int_{S} w d μ_{*}$ . Therefore $L_{w, K} μ_{*} = λ μ_{*}$ .

Conversely, suppose $L_{w, K} μ_{*} = λ μ_{*}$ . Since $μ_{*}$ has total mass 1, $L_{w, K} μ_{*} (S) = λ$ . Therefore

R_{w, K} (μ_{*}) = \frac{λ μ _{*}}{λ} = μ_{*} .

Thus evolutionary equilibrium is a Perron–Frobenius-type eigenproblem.

17Iteration

Proposition17.1Linearization of the orbit

Let $μ_{n + 1} = R_{w, K} (μ_{n})$ . When $w$ and $K$ are time-independent,

μ_{n} = \frac{L _{w, K}^{n} μ _{0}}{L _{w, K}^{n} μ _{0} ( S )} .

Proof.

Normalization factors are scalars and therefore cancel under successive projectivization. Induction gives the formula.

The nonlinear normalized dynamics is induced by an underlying positive linear dynamics.

18Perron–Frobenius theorem in the finite case

Let $A = W Q$ .

Theorem18.1Primitive finite mutation–selection dynamics

Suppose $A$ is primitive: there exists $k$ such that every entry of $A^{k}$ is strictly positive. Then:

$A$ has a unique positive dominant eigenvalue $λ_{PF} > 0$ .
There is a unique probability row vector $x_{*}$ with strictly positive components satisfying $x_{*} A = λ_{PF} x_{*}$ .
$x_{*}$ is the unique interior fixed point of the replicator–mutator map.
For every nonzero initial vector $x_{0} \geq 0$ ,
$\frac{x _{0} A ^{n}}{x _{0} A ^{n} 1} ⟶ x_{*} .$

Proof.

This is the Perron–Frobenius theorem for primitive nonnegative matrices applied to $A$ . The fixed-point characterization follows from Theorem 16.1. The convergence statement follows from domination of all other eigenvalues in modulus by $λ_{PF}$ .

Thus mutation plus selection can produce a unique asymptotic semantic population even though individual replication events remain stochastic.

19Continuous-time pure selection

Let the fitness over a short interval $h$ be

e^{h f_{i}} = 1 + h f_{i} + O (h^{2}) .

With no mutation,

x_{i} (t + h) = \frac{x _{i} ( t ) e ^{h f_{i}}}{j \sum x _{j} ( t ) e ^{h f_{j}}} .

Expanding,

x_{i} (t + h) = x_{i} + h x_{i} (f_{i} - f ˉ) + O (h^{2}),

where $f ˉ = \sum_{j} x_{j} f_{j}$ . Hence

x ˙_{i} = x_{i} (f_{i} - f ˉ) .

This is the continuous replicator equation.

20Mutation generated continuously

Let $L = (L_{ij})$ be a Markov generator: $L_{ij} \geq 0$ for $i \neq = j$ , and $\sum_{j} L_{ij} = 0$ . Set

Q (h) = I + h L + O (h^{2}), W (h) = I + h F + O (h^{2}),

where $F = diag (f_{i})$ . The finite-step update is

x (t + h) = \frac{x ( t ) W ( h ) Q ( h )}{x ( t ) W ( h ) 1} .

Theorem20.1Continuous selection–mutation equation

The infinitesimal evolution is

x ˙_{j} = x_{j} (f_{j} - f ˉ) + i \sum x_{i} L_{ij} .

Proof.

Expand

(I + h F) (I + h L) + O (h^{2}) = I + h (F + L) + O (h^{2}) .

The numerator is therefore

x + h (x F + xL) + O (h^{2}) .

Since $L 1 = 0$ , the denominator is $1 + h f ˉ + O (h^{2})$ . Using $(1 + h f ˉ)^{- 1} = 1 - h f ˉ + O (h^{2})$ gives

x + h (x F - f ˉ x + xL) + O (h^{2}) .

The $j$ -th component yields the stated equation.

21Mutation at reproduction

There is another standard continuous-time model. Suppose type $i$ reproduces at rate $f_{i}$ , and each offspring has type $j$ with probability $Q_{ij}$ . Then the production rate of type $j$ is $\sum_{i} x_{i} f_{i} Q_{ij}$ . Maintaining constant total population produces

x ˙_{j} = i \sum x_{i} f_{i} Q_{ij} - f ˉ x_{j},

where $f ˉ = \sum_{i} x_{i} f_{i}$ .

Proposition21.1Conservation of total probability

j \sum x ˙_{j} = 0.

Proof.

j \sum x ˙_{j} = i \sum x_{i} f_{i} (j \sum Q_{ij}) - f ˉ j \sum x_{j} = i \sum x_{i} f_{i} - f ˉ = 0.

22Semantic coarse-graining

Return now to observer-relative semantics. Suppose

q : X \to S

maps fine-grained memetic states to coarse semantic classes. Let $w : X \to R_{> 0}$ be microscopic fitness and

K : X \to P (X)

the microscopic mutation kernel. We ask when there is an induced closed evolutionary dynamics on $S$ . This is a categorical lumpability problem.

23Fitness compatibility

Definition23.1Fitness descends through q

Fitness is $q$ -observable if there exists $w : S \to R_{> 0}$ such that

w = w \circ q .

Thus all fine states in one semantic equivalence class possess the same fitness.

24Mutation compatibility

Definition24.1Kernel lumpability

The fine mutation kernel $K$ is $q$ -lumpable if there exists $K : S \to P (S)$ such that

P (q) \circ K ≃ K \circ q .

Diagrammatically,

X ↓ q S K K P (X) ↓ P (q) P (S) .

The probability distribution of the coarse offspring type therefore depends only on the coarse parent type.

25Coarse-graining theorem

Let $q_{*} : P (X) \to P (S)$ denote pushforward.

Theorem25.1Exact semantic coarse-graining commutes with evolution

Suppose

w = w \circ q and P (q) \circ K ≃ K \circ q .

Then

q_{*} \circ R_{w, K} ≃ R_{w, K} \circ q_{*} .

Proof.

Let $μ \in P (X)$ . First consider selection. For a coarse observable $f : S \to R$ ,

\int_{X} f d (q_{*} Sel_{w} (μ)) = \frac{\int _{X} f ( q ( x )) w ( x ) d μ ( x )}{\int _{X} w d μ} .

Since $w = w \circ q$ , the right-hand side is

\frac{\int _{S} f ( s ) w ( s ) d ( q _{*} μ ) ( s )}{\int _{S} w ( s ) d ( q _{*} μ ) ( s )} .

Hence

q_{*} Sel_{w} (μ) = Sel_{w} (q_{*} μ) .

Lumpability gives

q_{*} \circ K_{*} = K_{*} \circ q_{*} .

Therefore

q_{*} R_{w, K} (μ) = q_{*} K_{*} Sel_{w} (μ) = K_{*} q_{*} Sel_{w} (μ) = K_{*} Sel_{w} (q_{*} μ) = R_{w, K} (q_{*} μ) .

26Consequence for cognitive capability

Suppose $q_{c} : M \to S_{c}$ is the semantic localization associated with cognitive architecture $c$ . Then the population dynamics perceived by $c$ is autonomous precisely when both fitness and transition probabilities depend only on $q_{c} (m)$ . Otherwise the coarse semantic dynamics need not be Markovian.

Corollary26.1Hidden semantic structure can induce apparent memory

If mutation or fitness varies inside the fibers of $q : X \to S$ , then the coarse population process on $S$ generally fails to close as a first-order Markov process.

Two fine populations can have the same coarse distribution $q_{*} μ = q_{*} ν$ while having different distributions inside each fiber. If those hidden differences influence reproduction or mutation, then

q_{*} R (μ) \neq = q_{*} R (ν) .

The next coarse state therefore cannot be determined from the current coarse state alone. Thus insufficient cognitive resolution can convert microscopically Markovian dynamics into apparently non-Markovian dynamics.

27Observer-relative fitness

Fitness itself may depend on the observer or environment. For each architecture $c$ , let $w_{c} : S_{c} \to R_{> 0}$ . Then the same underlying memetic population may possess different induced evolutionary dynamics at different semantic resolutions.

For $u : c \to d$ , compatibility requires

w_{c} \circ u^{*} ≃ w_{d}

in the appropriate contravariant sense. Without this compatibility, there is no reason for selection to commute with change of observer.

28Replication probability

Let $[m]_{c}$ be a semantic equivalence class. A kernel $K$ defines a probability that state $m$ remains in its class after transmission.

Definition28.1Semantic fidelity

Define

F_{c} (m) = K (m, [m]_{c}) .

Thus $0 \leq F_{c} (m) \leq 1$ . If $c$ is a coarser observer than $d$ , then typically $[m]_{d} \subseteq [m]_{c}$ . Consequently

F_{c} (m) \geq F_{d} (m),

provided both classes are measured under the same underlying transmission kernel.

Proposition28.2Coarser semantics weakly increase measured fidelity

If $[m]_{d} \subseteq [m]_{c}$ , then $K (m, [m]_{d}) \leq K (m, [m]_{c})$ .

Proof.

Probability measures are monotone under inclusion of measurable events.

This is the stochastic analogue of the earlier theorem that coarse observers weakly increase apparent deterministic fidelity.

29Effective reproductive success

For a state $m$ , take $w (m)$ as exposure or reproductive intensity and $F_{c} (m)$ as semantic fidelity. A simple capability-relative effective self-reproduction factor is then

w (m) F_{c} (m) .

More generally, one should retain the full offspring kernel rather than compressing it to a scalar. Nevertheless, the factorization of reproductive intensity by semantic fidelity is useful conceptually.

30Selection of semantic classes

Let $A \subseteq S$ be a class of memes. The expected post-reproduction mass entering $A$ , before normalization, is

\int_{S} w (s) K (s, A) d μ (s) .

This decomposes evolutionary success into current prevalence, reproductive weight, and transition probability into the target class. A class can increase either because it reproduces rapidly or because other classes mutate into it.

31Mutation–selection balance

At equilibrium,

L_{w, K} μ_{*} = λ μ_{*} .

Hence for every measurable $A$ ,

λ μ_{*} (A) = \int_{S} w (s) K (s, A) d μ_{*} (s) .

This is the abstract mutation–selection balance equation. The quantity $λ$ is the asymptotic population growth factor before normalization.

32Multi-parent stochastic recombination

The Markov-kernel formalism also extends the operadic theory of Note II. Let $O (n)$ be an object of $n$ -ary operations. A stochastic operadic transformation is a kernel

K_{n} : O (n) \times S^{n} ⟶ P (S) .

Given parents $(s_{1}, \dots, s_{n})$ and an operation $o$ , the output is distributed according to $K_{n} (o; s_{1}, \dots, s_{n})$ .

Definition32.1Parent-relative stochastic fidelity

Relative to parent $j$ ,

K_{n} (o; s, [s_{j}]_{c})

is the probability that the recombination event preserves the semantic class of the $j$ -th parent. Deterministic operadic replication is recovered when the kernel is a Dirac distribution.

33Stochastic network semantics

More generally, replace deterministic network morphisms $X \to Y$ by stochastic morphisms $X \to P (Y)$ . Sequential network composition becomes Kleisli composition. Parallel composition is supplied by commutativity of the probability monad. Hence a probabilistic memetic network defines

N : X \to P (Y) .

Its semantics is a conditional probability law rather than a deterministic function.

34Replication relative to an observable

Let $Q : Y \to Z$ be a semantic observable. For an input $x$ , the output semantic distribution is $P (Q) (N (x))$ . If the target semantic class associated with $x$ is represented by $q (x) \in Z$ , then exact stochastic replication means

P (Q) (N (x)) = δ_{q (x)} .

Approximate replication permits a nonzero error distribution around this Dirac state.

35Error kernel

Define the semantic error kernel by

E_{Q} = P (Q) \circ N .

The deterministic theory corresponds to the special case in which $E_{Q} (x)$ is always a Dirac measure. The stochastic theory instead retains the entire distribution $E_{Q} (x) \in P (Z)$ . Thus mutation is not a Boolean event but a probability distribution over semantic deviations.

36Categorical evolutionary hierarchy

The successive theories now form the hierarchy

deterministic unary replication ↓ operadic recombination ↓ polynomial and network dynamics ↓ stochastic kernels ↓ population mutation-selection dynamics.

At each stage, the earlier theory is recovered by restriction. A deterministic morphism $f : X \to Y$ embeds into stochastic dynamics through the Dirac kernel $x \mapsto δ_{f (x)}$ .

37Full stochastic specification

The resulting model is the tuple

(C, J, H, P, M, Q, S, w, K) .

Symbol	Meaning
$C$	Cognitive architectures.
$J$	Coverage.
$H$	Semantic ∞-topos.
$P$	Probability monad.
$M$	Fine memetic realization object.
$Q : M \to S$	Semantic quotient.
$S$	Semantic class object.
$w$	Positive fitness.
$K$	Mutation or transmission kernel.

The evolutionary operator is

R_{w, K} (μ) = K_{*} \frac{w μ}{\int _{S} w d μ} .

38Main theorem

Theorem38.1Categorical replicator–mutator theorem

Let $H$ be a cognitive semantic ∞-topos equipped with a compatible probability monad $P$ . Let $S \in H$ , let $w : S \to R_{> 0}$ , and let $K : S \to P (S)$ . Then:

(i) Selection defines a normalization-preserving map $Sel_{w} : P (S) \to P (S)$ . See Proposition 6.2.
(ii) The Markov kernel induces a probability-preserving pushforward $K_{*}$ . See Proposition 8.1.
(iii) Their composite $K_{*} \circ Sel_{w}$ defines a discrete-time population dynamical system. See Definition 9.1.
(iv) This nonlinear system is the projectivization of the positive linear operator $L_{w, K}$ . See Corollary 11.3.
(v) Its stationary populations are precisely normalized positive eigenmeasures of $L_{w, K}$ . See Theorem 16.1.
(vi) For finite semantic state spaces it reduces to $x_{j}^{'} = \frac{\sum _{i} x _{i} w _{i} Q _{ij}}{\sum _{i} x _{i} w _{i}} .$ See Theorem 12.1.
(vii) Its infinitesimal pure-selection limit is $x ˙_{i} = x_{i} (f_{i} - f ˉ)$ . See Section 19.
(viii) With a continuous mutation generator $L$ , its infinitesimal limit is $x ˙_{j} = x_{j} (f_{j} - f ˉ) + \sum_{i} x_{i} L_{ij}$ . See Theorem 20.1.
(ix) A semantic quotient $q : X \to S$ inherits closed replicator–mutator dynamics whenever fitness is constant on its fibers and the microscopic mutation kernel is $q$ -lumpable. See Theorem 25.1.
(x) Failure of these conditions can make a coarse observer perceive non-Markovian dynamics even when the underlying dynamics is Markovian. See Corollary 26.1.

39Conceptual consequence

The deterministic theory characterized memetic replication as preservation of a semantic invariant:

Q \circ N ≃ Q \circ π_{j} .

The stochastic theory replaces this by a conditional distribution over semantic outcomes. Exact inheritance is Dirac-valued. Imperfect inheritance is a Markov kernel on semantic classes.

Population evolution is then obtained by weighting those states by reproductive success and integrating over all transition possibilities. The fundamental operator is

μ ⟼ \frac{K _{*} ( w μ )}{\int _{S} w d μ} .

40Final synthesis

The entire theory developed so far can now be compressed into the sequence

Realizations, then semantic classes, then selection, then mutation, then normalization, then population evolution.

At the individual level,

m ⇝ K (Q (m))

gives a distribution of possible semantic descendants. At the population level,

μ ⇝ R_{w, K} (μ) .

At equilibrium,

L_{w, K} μ_{*} = λ μ_{*} .

At finite dimension this is a Perron–Frobenius eigenproblem.

At different cognitive resolutions, exact coarse-graining is governed by the commuting square $P (q) \circ K ≃ K \circ q$ together with $w = w \circ q$ . Consequently, cognitive capability does not merely affect which individual transmissions count as faithful. It determines whether the population dynamics itself closes at the observer’s semantic resolution.

The strongest form of the theory is therefore

Normalized positive stochastic dynamics on observer-relative semantic classes.

Equivalently,

Projectivize (K_{*} \circ M_{w}) .

This connects the ∞-topos formulation of observer-relative identity directly to Markov kernels, mutation–selection systems, transfer operators, and the classical replicator–mutator equation.